How to move selected vertices on Z axis by 1mm
You can type in your transforms if you call the transform using direct shortcuts to begin with (G to move/grab, R to rotate, S to scale).

Just hit the key for your transform, here G, then the key of the axis you want to move in, here Z, then you can type the amount of transform you want and press &#9166; Enter.

Unfortunately, you can't write "1mm" and press &#9166; Enter: it only takes numbers and simple maths operations, it uses the units set in your scene settings. 

Blender is set to meters by default, so for moving by one millimeter you will have to write 0.001 (with a dot, not a comma) and hit &#9166; Enter.
You can also just type in the new global or local coordinates of your object in the Transform input in your viewport's sidebar. You can even let Blender make the maths operations for your by putting your cursor at the end of the value you want to edit and write simple maths like +0.01 or -1, ...

TL;DR type GZ 0.001 and &#9166; Enter.

If you will often need to type in millimeters, then it will be faster for you to set the units to milimeters so you don't have to convert millimeters to meters all the time.

Head over to the properties editor, Scene tab (white triangle icon), Units tab, and change the Length unit to millimeters. This just changes how units are displayed, it does'nt change any scale internally.